I don't know which sort of bandsaw you have but I have one of the standard Chinese type that are rated as doing up to 4.5 by 6 inches rectangular, or about 4 inches round. I have cut much larger sizes by means of a couple of bodges. The sensitive should avert their eyes about now!
Ok, for doing about six inches diameter, you can cut until the saw frame is just fouling the job, then turn the job in the vice to cut through the rest. Obviously the cut must remain aligned with the saw.
For an even larger job, eight inches diameter, I have managed to set things up with the bulk of the stock out to the right of the saw, and only the slice being cut off on the side with the frame. So long as the slice is not thick enough to foul the frame itself this will work fine. It does mean coming up with a way of holding the stock so that it does not move relative to the saw.
For cutting pieces where the stock is too short for the vice, you can sometimes get enough grip by putting a piece of similar thickness at the left end of the vice so that the jaws remain parallel.
These sort of tricks only work because the cutting loads with a bandsaw are quite low. Don't try to get too tricky with a reciprocating saw.
John